Nonlinear penetrative convection

Abstract

Convection in water above ice penetrates into the stably stratified region above the density maximum at 4 °C. Two-dimensional penetrative convection in a Boussinesq fluid confined between free boundaries has been studied in a series of numerical experiments. These included cases with a constant temperature at both boundaries as well as cases with a fixed average flux at the lower boundary. Steady convection occurs at Rayleigh numbers below the critical value predicted by linear theory. At high Rayleigh numbers, resonant coupling between convection and gravitational modes in the stable layer excites finite amplitude oscillations. The problem can be described by a simplified model which allows for distortion of the mean temperature profile and balances the convected and conducted flux. This model explains the finite amplitude instability and predicts the Nusselt number as a function of Rayleigh number. These predictions are in excellent agreement with the computed results.